Options Trading Formulas

The cornerstone of options pricing. Calculates the theoretical fair value of a European call option based on five inputs.

C = S · N(d₁) - K · e⁻ʳᵀ · N(d₂)

Calculates the theoretical fair value of a European put option. Mirror image of the call formula using negative d values.

P = K · e⁻ʳᵀ · N(-d₂) - S · N(-d₁)

The two critical intermediate values in Black-Scholes. d₁ measures moneyness adjusted for drift, d₂ adjusts for volatility over time.

d₁ = [ln(S/K) + (r + σ²/2) · T] / (σ · √T)
d₂ = d₁ - σ · √T

The standard normal probability density function — the famous bell curve. Used heavily in calculating Greeks like Gamma and Vega.

φ(x) = (1 / √(2π)) · e⁻ˣ²ᐟ²

The real, tangible value of an option if exercised immediately. Only in-the-money options have intrinsic value.

IV_call = max(S - K, 0)
IV_put  = max(K - S, 0)

The portion of an option's price above its intrinsic value. Represents the premium paid for time and uncertainty — this is what theta decays.

Extrinsic = Option Price - Intrinsic Value

A simpler, more intuitive alternative to Black-Scholes. Prices options by modeling the stock as moving up or down by a fixed factor each period.

C = [p · Cᵤ + (1-p) · Cᵈ] / (1+r)
p = (1+r - d) / (u - d)

The first-order price sensitivity. Measures how much the option price changes for a $1 move in the underlying stock.

Δ_call = N(d₁)
Δ_put  = N(d₁) - 1

The second-order price sensitivity. Measures how fast Delta changes as the stock moves — it's the "acceleration" of the option.

Γ = φ(d₁) / (S · σ · √T)

Time decay — the silent killer of long options. Measures how much option value erodes each day, all else being equal.

Θ_call = [-(S · φ(d₁) · σ) / (2√T) - r · K · e⁻ʳᵀ · N(d₂)] / 365
Θ_put  = [-(S · φ(d₁) · σ) / (2√T) + r · K · e⁻ʳᵀ · N(-d₂)] / 365

Volatility sensitivity. Measures how much the option price changes when implied volatility moves by 1 percentage point.

ν = S · φ(d₁) · √T / 100

Interest rate sensitivity. The least-used Greek for short-term traders, but significant for LEAPS and in high-rate environments.

ρ_call = K · T · e⁻ʳᵀ · N(d₂) / 100
ρ_put  = -K · T · e⁻ʳᵀ · N(-d₂) / 100

Second-order Greek measuring how Delta changes as time passes. Critical for understanding why your delta-hedged position drifts overnight.

Charm = -φ(d₁) · [2(r-q)T - d₂·σ·√T] / (2T·σ·√T)

Cross-Greek measuring how Delta changes when IV changes (or equivalently, how Vega changes when the stock moves). Key driver of "vol smile dynamics."

Vanna = -φ(d₁) · d₂ / (S · σ · √T)

Second-order volatility Greek. Measures how Vega itself changes when implied volatility moves. The "Gamma of Vega."

Vomma = ν · (d₁ · d₂) / σ

Third-order price Greek. Measures how Gamma changes as the stock moves. Important for understanding gamma risk on large portfolio moves.

Speed = -Γ · (d₁ + σ√T) / (S · σ · √T)

Dividend yield sensitivity. Measures how much the option price changes when the continuous dividend yield changes by 1%. Critical for high-dividend stocks and indices.

ε_call = -T · S · e⁻ᵠᵀ · N(d₁) / 100
ε_put  = T · S · e⁻ᵠᵀ · N(-d₁) / 100

Third-order Greek measuring how Gamma changes when volatility changes. Important for volatility traders who need to understand gamma risk across different vol regimes.

Zomma = Γ · (d₁ · d₂ - 1) / σ

Measures the actual, realized price fluctuations of a stock over a past period. Also called "realized volatility" or "statistical volatility."

HV = σ_daily × √252
σ_daily = StdDev(ln(Sᵢ / Sᵢ₋₁))

Where is current IV relative to its 52-week range? Tells you if vol is high or low in a simple 0-100% scale.

IVR = (IV_current - IV_52w_low) / (IV_52w_high - IV_52w_low) × 100

What percentage of days in the past year had lower IV than today? More robust than IV Rank because it accounts for time spent at each level.

IVP = (Days with IV < IV_current) / Total Trading Days × 100

Converts the VIX (annualized expected volatility of S&P 500) to an expected daily percentage move.

Daily Move = VIX / √252

Implied volatility cannot be solved algebraically — it must be found numerically. The Newton-Raphson method iteratively converges on the IV that makes Black-Scholes match the market price.

σ_{n+1} = σ_n - (BS(σ_n) - Market Price) / Vega(σ_n)

A historical volatility estimator using high-low prices instead of closing prices. Often more accurate than close-to-close HV because it uses intraday range information.

σ_P = √[(1 / (n × ln 2)) × Σ(ln(Hᵢ / Lᵢ))²]
Annualized = σ_P × √252

Combines close-to-close, high-low, and open-close data for the most efficient historical volatility estimator. Uses all available price information.

σ² = (0.5/n) × Σ(ln(Hᵢ/Lᵢ))² - (2ln2-1)/n × Σ[ln(Cᵢ/Cᵢ₋₁)]²
Annualized = √(σ² × 252)

Defines the risk/reward for bull/bear call or put spreads. The most common defined-risk options strategy.

Max Profit (credit) = Premium Received
Max Loss (credit) = Width - Premium Received
Breakeven = Short Strike ± Premium

The classic neutral income strategy: sell a put spread and a call spread simultaneously. Profit if the stock stays within a range.

Max Profit = Total Credit Received
Max Loss = Wider Wing Width - Total Credit
BE_low = Short Put - Credit
BE_high = Short Call + Credit

A straddle (buy/sell ATM call + ATM put) profits from big moves (long) or stillness (short). Breakevens define the required move size.

Long Straddle:
BE_up = Strike + Total Premium Paid
BE_down = Strike - Total Premium Paid

A strangle uses OTM options instead of ATM, requiring a bigger move but costing less premium.

Long Strangle:
BE_up = Call Strike + Total Premium
BE_down = Put Strike - Total Premium

Selling calls against stock you own to generate income. The most popular "first options strategy" and the basis of many income portfolios.

Max Return = (Premium + (K - S)) / S × 100
BE = S - Premium
Annualized = Return × (365 / DTE)

A low-cost, pinning strategy that profits when the stock expires near a specific price. Buy 1 lower + buy 1 upper + sell 2 middle.

Max Profit = Wing Width - Net Debit
Max Loss = Net Debit Paid
BE_low = Lower Strike + Debit
BE_high = Upper Strike - Debit

Buy a longer-dated option and sell a shorter-dated option at the same strike. Profits from the front-month decaying faster (theta differential).

Max Profit ≈ at expiration of front month, stock at strike
Max Loss = Net Debit Paid
Cost = Long Option - Short Option

Sell more options than you buy to collect net credit. Profits when the stock moves toward your short strikes but not too far.

Max Profit = Premium Received + (Width × Long Qty)
Max Loss (call side) = Unlimited
Max Loss (put side) = Width × (Short Qty - Long Qty) - Credit

Combine different strikes AND different expirations. The best of both worlds: directional exposure plus theta decay differential.

Max Profit ≈ at front expiry, stock at long strike
Max Loss = Net Debit Paid
Cost = Long Back Month - Short Front Month

An arbitrage strategy combining bull call spread and bear put spread. Creates a synthetic loan or borrow depending on pricing.

Box Value = |K₁ - K₂| × e⁻ʳᵀ
Profit = Box Value - Net Premium Paid
Arbitrage if: Net Premium < Box Value

The fundamental relationship linking European calls, puts, stock, and bonds. If violated, an arbitrage opportunity exists.

C - P = S - K · e⁻ʳᵀ

Replicate a stock position using only options. Useful when you can't or don't want to trade the stock directly (margin efficiency, short-sale restrictions).

Synthetic Long  = +Call - Put (same strike & expiry)
Synthetic Short = -Call + Put (same strike & expiry)

Arbitrage strategies that exploit put-call parity violations. A conversion locks in profit when options are mispriced relative to the stock.

Conversion: +Stock, +Put, -Call (same strike)
Profit = C - P - (S - K·e⁻ʳᵀ)

Quick estimation of how likely a trade is profitable at expiration. Based on Delta for single legs, or the net premium and strikes for spreads.

POP_credit = 1 - Δ_short_strike
POP_debit  = Δ_long_strike

Predicts the stock's likely price range by expiration based on implied volatility. The "1-sigma" move represents 68% confidence.

EM = S × IV × √(DTE / 365)

The mathematical expected profit/loss of a trade considering all possible outcomes weighted by their probabilities. Positive EV = edge.

EV = (POP × Avg Win) - ((1-POP) × Avg Loss)

Maps standard deviation multiples to probability ranges. Tells you the chance of the stock staying within N standard deviations.

1σ ≈ 68.2%, 2σ ≈ 95.4%, 3σ ≈ 99.7%
Range = S ± N × S × IV × √(DTE/365)

The probability that the stock will touch a given strike at any point before expiration — roughly double the probability of expiring ITM.

P(touch) ≈ 2 × Δ_strike

Delta is approximately equal to the risk-neutral probability of finishing in-the-money. A quick way to estimate probabilities without complex calculations.

P(ITM) ≈ |Δ|
For OTM options: use |Δ|
For ITM options: P(ITM) = 1 - |Δ_short_counterpart|

The most important formula in trading. Determines how many contracts to trade based on your account size and maximum acceptable loss per trade.

Contracts = (Account × Risk%) / Max Loss per Contract

The mathematically optimal bet size that maximizes long-term growth rate. In practice, traders use "half Kelly" or less for safety.

f* = (p · b - q) / b
where q = 1 - p

The net directional exposure of your entire portfolio. Positive = bullish, negative = bearish, zero = neutral.

Portfolio Δ = Σ (Δᵢ × Quantityᵢ × 100)

Normalizes all positions to SPY-equivalent delta. Allows comparing apples-to-apples exposure across stocks with different volatilities.

BWΔ = Position Δ × (Stock Price / SPY Price) × Beta

Measures the potential return relative to the capital at risk (buying power reduction). The primary metric for comparing options strategies.

ROC = (Credit Received / Buying Power) × 100
Annualized ROC = ROC × (365 / DTE)

Measures risk-adjusted returns — how much return you earn per unit of risk taken. The gold standard for comparing trading strategies.

Sharpe = (R_p - R_f) / σ_p

The largest peak-to-trough decline in your account. The single most important metric for knowing if a strategy is survivable.

MDD = (Peak Value - Trough Value) / Peak Value × 100
Recovery = 1 / (1 - MDD%) - 1

Improved Sharpe ratio that only penalizes downside volatility, not upside. More appropriate for strategies with asymmetric returns (like selling premium).

Sortino = (R_p - R_f) / σ_down
σ_down = √[Σ min(0, R_i - R_f)² / n]

Estimates maximum expected loss at a given confidence level over a time period, assuming normal distribution of returns.

VaR = Position × (μ - z × σ × √t)
where z = 1.65 (95%), 2.33 (99%)

Measures how positions move together. Critical for understanding true portfolio diversification beyond just counting different tickers.

ρ = Σ[(xᵢ - x̄)(yᵢ - ȳ)] / (√Σ(xᵢ - x̄)² × √Σ(yᵢ - ȳ)²)